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On the equality of the BLUPs under two linear mixed models. (English) Zbl 1226.62066
Summary: We consider two mixed linear models, \({\mathcal{M}_1}\) and \({\mathcal{M}_2}\), say, which have different covariance matrices. We review some useful concepts and results on the best linear unbiased estimators (BLUEs) and on best linear unbiased predictors (BLUPs). We give new necessary and sufficient conditions, without making any rank assumptions, that every representation of the BLUP of the random effect under the model \({\mathcal{M}_1}\) continues to be BLUP under the model \({\mathcal{M}_2}\). These considerations are generalized to two linear models with new unobserved future observations.

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
Full Text: DOI
[1] Baksalary JK (2004) An elementary development of the equation characterizing best linear unbiased estimators. Linear Algebra Appl 388: 3–6 · Zbl 1052.62062 · doi:10.1016/S0024-3795(03)00396-3
[2] Christensen R (2002) Plane answers to complex questions: the theory of linear models. 3. Springer, New York · Zbl 0992.62059
[3] Goldberger AS (1962) Best linear unbiased prediction in the generalized linear regression model. J Am Stat Assoc 58: 369–375 · Zbl 0124.35502 · doi:10.1080/01621459.1962.10480665
[4] Haslett J, Haslett SJ (2007) The three basic types of residuals for a linear model. Int Stat Rev 75: 1–24 · doi:10.1111/j.1751-5823.2006.00001.x
[5] Haslett SJ, Puntanen S (2010) Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat Pap 51. doi: 10.1007/s00362-009-0219-7 · Zbl 1247.62167
[6] Henderson CR (1950) Estimation of genetic parameters. Ann Math Stat 21: 309–310
[7] Henderson CR (1963) Selection index and expected genetic advance. Statistical Genetics and Plant Breeding, National Research Council Publication No. 982. National Academy of Sciences, Washington, pp 141–163
[8] Isotalo J, Puntanen S (2006) Linear prediction sufficiency for new observations in the general Gauss–Markov model. Commun Stat Theory Methods 35: 1011–1023 · Zbl 1102.62072 · doi:10.1080/03610920600672146
[9] Isotalo J, Möls M, Puntanen S (2006) Invariance of the BLUE under the linear fixed and mixed effects models. Acta et Commentationes Universitatis Tartuensis de Mathematica 10: 69–76 · Zbl 1136.62043
[10] Isotalo J, Puntanen S, Styan GPH (2008) A useful matrix decomposition and its statistical applications in linear regression. Commun Stat Theory Methods 37: 1436–1457 · Zbl 1163.62051 · doi:10.1080/03610920701666328
[11] Kala R (1981) Projectors and linear estimation in general linear models. Commun Stat Theory Methods 10: 849–873 · Zbl 0465.62060 · doi:10.1080/03610928108828078
[12] Mitra SK, Moore BJ (1973) Gauss–Markov estimation with an incorrect dispersion matrix. Sankhyā, Ser A 35: 139–152 · Zbl 0277.62044
[13] Möls M (2004) Linear mixed models with equivalent predictors. PhD Thesis. Dissertationes Mathematicae Universitatis Tartuensis, 36 · Zbl 1165.62328
[14] Puntanen S, Styan GPH (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator (with discussion). The American Statistician 43:151–161 (Commented by O. Kempthorne on pp. 161–162 and by S. R. Searle on pp. 162–163, Reply by the authors on p. 164)
[15] Rao CR (1967) Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Le Cam LM, Neyman J (eds) Proceedings of the fifth Berkeley symposium on mathematical statistics and probability: Berkeley, California, 1965/1966, vol. 1, University of California Press, Berkeley, pp 355–372
[16] Rao CR (1968) A note on a previous lemma in the theory of least squares and some further results. Sankhyā, Ser A 30: 245–252
[17] Rao CR (1971) Unified theory of linear estimation. Sankhyā, Ser A 33:371–394. (Corrigendum (1972), 34, p. 194 and p. 477) · Zbl 0236.62048
[18] Rao CR (1973) Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J Multivar Anal 3: 276–292 · Zbl 0276.62068 · doi:10.1016/0047-259X(73)90042-0
[19] Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York · Zbl 0236.15004
[20] Rao CR, Toutenburg H, Shalabh, Heumann C (2008) Linear models and generalizations: least squares and alternatives. 3. Springer, New York · Zbl 1151.62063
[21] Robinson GK (1991) That BLUP is a good thing: the estimation of random effects (with discussion on pp. 32–51). Stat Sci 6: 15–51 · Zbl 0955.62500 · doi:10.1214/ss/1177011926
[22] Roy A (2008) Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up. J Appl Stat 35: 307–320 · Zbl 1147.62052 · doi:10.1080/02664760701833271
[23] Searle SR (1997) The matrix handling of BLUE and BLUP in the mixed linear model. Linear Algebra Appl 264: 291–311 · Zbl 0889.62059 · doi:10.1016/S0024-3795(96)00400-4
[24] Zyskind G (1967) On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. Ann Math Stat 38: 1092–1109 · Zbl 0171.17103 · doi:10.1214/aoms/1177698779
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